An optimization algorithm may be used to determine a solution for a problem in which the solution considers all variables and constraints related to the problem and provides a lowest cost configuration of the values for all the variables. For example, in a televised event, a plurality of cameras may be used to capture the event. The cameras may move along planned trajectories to fully capture the event. Accordingly, the problem associated with such a scenario may be to determine the planned trajectories for the cameras that incorporate the variables and constraints associated therewith. Such variables and constraints may include no overlap of space as the cameras cannot physically be co-located at a common time, a trajectory path that provides a sufficient video capture of the event, a tracking algorithm for the occurrences during the event, etc. Through incorporation of all these considerations, the optimization algorithm may determine the trajectories of the cameras to provide the lowest cost solution (e.g., least distance to be covered by the cameras, best coverage of the event, least energy requirements, etc.). However, most optimization algorithms require significantly high processing requirements for all these considerations to be assessed as well as provide the lowest cost solution to the problem.
A conventional algorithm used for convex optimization is the Alternating Direction Method of Multipliers (ADMM). Conventionally, the ADMM algorithm is a variant of an augmented Lagrangian scheme that uses partial updates for dual variables. This algorithm has also been utilized and shown to be well-suited for distributed implementations. Further developments in using the ADMM algorithm have included being used in applications such as Linear Programming (LP) decoding of error-correcting codes and compressed sensing. However, the ADMM algorithm has in practice been used exclusively for convex optimization problems whereas determining optimal trajectories is not convex. Another conventional approach for optimization uses graphical models in which a graph denotes conditional dependencies between random variables. The graphical models often are combined with a Belief Propagation (BP) message-passing algorithm in which inferences are performed on the graphical models. The BP message-passing algorithm calculates the marginal distribution for each unobserved node, conditional on any observed nodes within the graphical model.
Even with altering the above conventional approaches to fit the optimization problem of determining trajectories, a further consideration in determining optimal trajectories involves the number of dimensions in which the objects may move. For example, the cameras may be limited to a surface upon which the camera is placed upon such as mounted on a land vehicle, placed on rails or tracks, etc. That is, this may represent a two-dimensional path restriction as they are limited to move along this plane. In another example, the cameras may be free to move in any direction which may represent a three-dimensional path availability such as aerial movements (e.g., mounted on an unmanned aerial vehicle) or underwater movements (e.g., mounted on an autonomous underwater vehicle). Therefore, even if the ADMM and/or BP algorithms are altered for use in the less complex two-dimensional analysis of determining optimal trajectories, the applicability of these altered algorithms will have limited applicability to the three-dimensional analysis.